Module 5 - Quizzes/Test
Use the given information to find the p-value. Also, use a 0.05 significance level and state the conclusion about the null hypothesis (reject the null hypothesis or fail to reject the null hypothesis). With H1: p≠0.377, the test statistic is z=3.06.
0.0022; reject the null hypothesis Use normalcdf in calc to find p-value (area) of given z # then multiple by 2 since hint: p≠0.377 ; meaning ≠ equates to 2 tailed test (which requires the found area to be multiplies by 2) and final p-value is lower than 0.05 significance level means H0 is rejected
Use the given information to find the p-value. Also, use a 0.05 significance level and state the conclusion about the null hypothesis (reject the null hypothesis or fail to reject the null hypothesis). With H1: p>0.554, the test statistic is z=1.34.
0.0901; fail to reject the null hypothesis - Use normalcdf function in calculator to find p-value (area) to the left of z=1.34 (Hint: p > 0.554)
Use the given information to find the p-value. Also, use a 0.05 significance level and state the conclusion about the null hypothesis (reject the null hypothesis or fail to reject the null hypothesis). The test statistic in a right-tailed test is z=0.52.
0.3015; fail to reject the null hypothesis Use normalcdf in calc to find p-value (area) of given z # This fail to reject b/c P-value > given α (0.05)
John performed a one-sample z-test for proportions and obtained a p-value of 0.35. John decided to reject the null hypothesis. What is the probability John made a Type I Error?
0.35 If the null hypothesis is rejected, the probability of incorrectly rejecting the null hypothesis is the p-value if no significance level is given. Note how high the p-value is. This is a reason why we do not want to reject the null hypothesis for high p-values, the probability of making a Type I Error would be quite large. The probability of making a Type I Error when rejecting the null hypothesis is the significance level if given or the p-value if a significance level is not given.
Cameron wondered if the average score on a final exam was different between those who texted on a regular basis during lecture for a particular class and those that did not text at all during lecture for the same class. Which of the following is the correct statement of what a Type II Error is in the context of this problem?
Cameron did not have evidence that there was a difference in the average scores on the final exam between those who texted during lecture and those who did not text during lecture when there was a difference in the average scores. - A Type II Error is when the null hypothesis is not rejected, but in fact the null hypothesis is false.
Dora likes to explore. Recently, she explored southeast Australia where she found some very large trees known as Eucalyptus regnans or Mountain Ash. She wondered if they were taller, on average, than the coastal Douglas Firs of her native state of Oregon in the United States, which have an average height of 250 feet in old growth areas. Dora measured the heights of 15 Mountain Ash trees in southeast Australia and found an average height of these trees of 293 feet. Suppose s=25 feet. Assume the heights of the 15 trees in Dora's sample are representative of the heights of all Mountain Ash trees in southeast Australia. State the null hypothesis.
H0: The average height of all Mountain Ash trees in southeast Australia is 250feet, the same as the average height of typical old growth Douglas Fir.
Express the null hypothesis and the alternative hypothesis in symbolic form. Use the correct symbol (μ, ρ, σ) for the indicated parameter. An entomologist writes an article in a scientific journal which claims that fewer than 16 in ten thousand male fireflies are unable to produce light due to a genetic mutation. Use the parameter p, the true proportion of fireflies unable to produce light.
H0: p=0.0016 H1: p<0.0016
Suppose 213 subjects are treated with a drug that is used to treat pain and 51 of them developed nausea. Use a 0.05 significance level to test the claim that more than 20% of users develop nausea. Identify the null and alternative hypotheses for this test. Choose the correct answer below.
H0: p=0.20 H1: p>0.20
Express the null hypothesis and the alternative hypothesis in symbolic form. Use the correct symbol (μ, ρ, σ) for the indicated parameter. A psychologist claims that more than 5.8% of the population suffers from professional problems due to extreme shyness. Use p, the true percentage of the population that suffers from extreme shyness.
H0: p=5.8% H1: p>5.8%
Express the null hypothesis and the alternative hypothesis in symbolic form. Use the correct symbol (μ, p, σ) for the indicated parameter. The manufacturer of a refrigerator system for beer kegs produces refrigerators that are supposed to maintain a true mean temperature, μ, of 48°F, ideal for a certain type of German pilsner. The owner of the brewery does not agree with the refrigerator manufacturer, and claims he can prove that the true mean temperature is incorrect.
H0: μ=48°F H1: μ≠48°F
A formal hypothesis test is to be conducted using the claim that the mean body temperature is equal to 98.6°F. What is the null hypothesis and how is it denoted?
H0: μ=98.6°F
Dora likes to explore. Recently, she explored southeast Australia where she found some very large trees known as Eucalyptus regnans or Mountain Ash. She wondered if they were taller, on average, than the coastal Douglas Firs of her native state of Oregon in the United States, which have an average height of 250 feet in old growth areas. Dora measured the heights of 15 Mountain Ash trees in southeast Australia and found an average height of these trees of 293 feet. Suppose s=25 feet. Assume the heights of the 15 trees in Dora's sample are representative of the heights of all Mountain Ash trees in southeast Australia. State the alternative hypothesis.
HA: μx>250 -In words,HA: the average height of the Mountain Ash trees in southeast Australia is more than 250 feet, the average height of Douglas Fir trees in Oregon.
Typically, the idea of the _______ hypothesis is that of "no effect," "no difference," or "no change."
Null
The _________ hypothesis is a statement that the value of a population parameter is equal to some claimed value.
Null The null hypothesis (denoted by H0) is a statement that the value of a population parameter (such as proportion, mean, or standard deviation) is equal to some claimed value.
Allison randomly sampled 40 students in a particular class and asked each how many hours per week they studied. She calculated a sample mean, x, of 9.5 hours per week. Her hypotheses were H0: μx=8 hours per week and HA: μx>8 hours per week, where x is the number of hours students spent studying per week. She performed a hypothesis test and obtained a p-value of 0.0003. What does this mean?
Of all the random samples of 40 students from this class, 0.03% would give a sample mean of 9.5 hours per week or more given that the true mean number of hours spent studying for this class for all students in the class is 8 hours per week. - The probability of observing the sample mean or something more unusual just by chance if the null hypothesis is true is the p-value.
The ________ is the probability of getting a test statistic at least as extreme as the one representing the sample data, assuming that the null hypothesis is true.
P-value
Assume a significance level of α=0.01 and use the given information to complete parts (a) and (b) below. Original claim: The mean pulse rate (in beats per minute) of a certain group of adult males is 69 bpm. The hypothesis test results in a P-value of 0.0068. State a conclusion about the null hypothesis. (Reject H0 or fail to reject H0.) Choose the correct answer below. Without using technical terms, state a final conclusion that addresses the original claim. Which of the following is the correct conclusion?
Reject H0 because the P-value is less than or equal to α. There is sufficient evidence to warrant rejection of the claim that the mean pulse rate (in beats per minute) of the group of adult males is 69 bpm.
Researchers wondered if the average water temperature of streams without tailed frogs was different than the average water temperature of streams with tailed frogs, which is known to be 12.2 degrees Celsius. A survey of 31 streams without tailed frogs was taken and the water temperature was recorded for each stream. Which of the following is the correct statement of what a Type I Error is in the context of this problem?
Researchers found evidence that the average water temperature of streams without tailed frogs was different than the average water temperature of streams with tailed frogs when there was no difference in the average temperatures. - A Type I Error happens when there is evidence to reject the null hypothesis, but in fact the null hypothesis is actually true.
Which of the following is a requirement for testing a claim about a population proportion?
The conditions np≥5 and nq≥5 are both satisfied.
Is the nutritional information listed on food items accurate? Researchers randomly sampled 12 frozen dinners of a certain type from production during a particular period. The calorie content was determined. The stated calorie content on the package was 240 calories. Researchers wanted to determine if there was evidence to indicate that the mean calorie count was not equal to 240 calories using the one-sample t-methods. Which of the following statements is true?
The distribution of sample means will be normal only if the population data (that is, calorie content of all frozen dinners of this type produced during this particular period) follow a normal distribution.
A medical study was investigating if getting a flu shot actually reduced the risk of developing the flu. From a group of adult volunteers, researchers randomly assigned half to receive an injection that contained the drug believed to reduce the risk of getting the flu and the other half to receive an injection containing no active ingredient (i.e. sugar water). A hypothesis test was performed and a p-value of 0.0002 was obtained. Which of the following statements is true?
The small p-value indicates strong evidence to reject the null hypothesis. Because an experiment was performed, it can be concluded that the reduction in the risk of getting the flu was caused by the flu shot. - In a properly designed experiment, if a difference in the response variable exists between the two groups being compared, that difference must be due to the treatment since all other factors (both known and unknown) should be balanced by the randomization. OK
Dora likes to explore. Recently, she explored southeast Australia where she found some very large trees known as Eucalyptus regnans or Mountain Ash. She wondered if they were taller, on average, than the coastal Douglas Firs of her native state of Oregon in the United States, which have an average height of 250 feet in old growth areas. Dora measured the heights of 15 Mountain Ash trees in southeast Australia and found an average height of these trees of 293 feet. Suppose s = 25 feet. Assume the heights of the 15 trees in Dora's sample are representative of the heights of all Mountain Ash trees in southeast Australia. The t-statistic for this problem is 6.661. Based on this t-statistic, what is the conclusion?
There is sufficient evidence to indicate that the average height of all Mountain Ash trees in southeast Australia is more than 250 feet. - The large t-statistic leads to a small p-value (0.00001). With such a p-value, we can use the adjective "strong" to describe the amount of evidence to reject the null hypothesis and say that the alternative hypothesis is true.
Formulate the indicated conclusion in nontechnical terms. Be sure to address the original claim. Carter Motor Company claims that its new sedan, the Libra, will average better than 32 miles per gallon in the city. Assuming that a hypothesis test of the claim has been conducted and that the conclusion is to reject the null hypothesis, state the conclusion in nontechnical terms.
There is sufficient evidence to support the claim that the mean is greater than 32 miles per gallon.
The Ericsson method is one of several methods claimed to increase the likelihood of a baby girl. In a clinical trial, results could be analyzed with a formal hypothesis test with the alternative hypothesis of p>0.5, which corresponds to the claim that the method increases the likelihood of having a girl, so that the proportion of girls is greater than 0.5. If you have an interest in establishing the success of the method, which of the following P-values would you prefer: 0.999, 0.5, 0.95, 0.05, 0.01, 0.001? Why?
The P-value of 0.001 is preferred because it corresponds to the sample evidence that most strongly supports the alternative hypothesis that the method is effective In a hypothesis test, the P-value is the probability of getting a value of the test statistic that is at least extreme as the test statistic obtained from the sample data, assuming that the null hypothesis is true. ** We are interested in the success of the method and thus are interested in supporting the alternative hypothesis of p > 0.5, which corresponds w/ rejecting the H0 p = 0.5 Meaning, if the P-value is less than significance level a, then we reject the H0, The smallest given P-value is what we would want in order to reject H0
Which of the following is NOT true about P-values in hypothesis testing?
The P-value separates the critical region from the values that do not lead to rejection of the null hypothesis. The P-value does not separate the critical region from the values that do not lead to rejection of the null hypothesis. The P-value is not a value on the horizontal axis, it is an area.
Identify the type I error and the type II error that correspond to the given hypothesis. The percentage of households with more than 1 pet is equal to 65%.
Type 1 Error: Reject the null hypothesis that the percentage of households with more than 1 pet is equal to 65% when that percentage is actually equal to 65%. Type 2 Error: Fail to reject the null hypothesis that the percentage of households with more than 1 pet is equal to 65% when that percentage is actually different from 65%.
When is a t-test performed instead of a z-test?
When the population standard deviation is not known
Dora likes to explore. Recently, she explored southeast Australia where she found some very large trees known as Eucalyptus regnans or Mountain Ash. She wondered if they were taller, on average, than the coastal Douglas Firs of her native state of Oregon in the United States, which have an average height of 250 feet in old growth areas. Dora measured the heights of 15 Mountain Ash trees in southeast Australia and found an average height of these trees of 293 feet. Suppose s=25 feet. Assume the heights of the 15 trees in Dora's sample are representative of the heights of all Mountain Ash trees in southeast Australia. The t-statistic for this problem is 6.661. Based on this t-statistic, which of the following is true?
With a p-value less than 0.0001, there is sufficient evidence to reject the null hypothesis and accept the alternative as true. - The smaller the p-value, the more evidence to reject the null hypothesis.
A certain drug is used to treat asthma. In a clinical trial of the drug, 21 of 271 treated subjects experienced headaches (based on data from the manufacturer). The accompanying calculator display shows results from a test of the claim that less than 9% of treated subjects experienced headaches. Use the normal distribution as an approximation to the binomial distribution and assume a 0.01 significance level to complete parts (a) through (e) below. 1-PropZTest prop<0.09 z= -0.7195700 p= 0.23589 p^= 0.0774907 n= 271
a. Is the test two-tailed, left-tailed, or right-tailed? - Left-tailed test b. What is the test statistic? - z= - 0.72 c. What is the P-value? - P-value = 0.2359 d. What is the null hypothesis, and what do you conclude about it? Identify the null hypothesis. - H0: p= 0.09 Decide whether to reject the null hypothesis. Choose the correct answer below. - Fail to reject the null hypothesis because the P-value is greater than the significance level, α e. What is the final conclusion? - There is not sufficient evidence to support the claim that less than 9% of treated subjects experienced headaches.
The test statistic of z=−3.13 is obtained when testing the claim that p=1/4. a. Using a significance level of α=0.05, find the critical value(s). b. Should we reject H0 or should we fail to reject H0?
a. The critical value(s) is/are z=negative 1.96 comma 1.96 or (−1.96,1.96) Find Z value when area= 0.05 (two tail) Use invNorm enter α=0.05/2 as area and find value, since the claim is wanting a value p=1/4, we want to include 2 values left and right sides (both tails) b. Reject H0. There is sufficient evidence to warrant rejection of the claim that p=1/4. H0 rejected because z value −3.13 is in the critical region to the left (Hint: p=1/4) of -1.96
The test statistic of z=−3.44 is obtained when testing the claim that p<0.33. a. Using a significance level of α=0.01, find the critical value(s). b. Should we reject H0 or should we fail to reject H0?
a. The critical value(s) is/are z=negative 2.33 or (−2.33). Find Z value when area= 0.01 (one tail) Use invNorm enter α=0.01 as area and find value, since the claim is wanting a value p<0.33, we only need to include 1 value to the left (left tail) b. Reject H0. There is sufficient evidence to support the claim that p<0.33. H0 rejected because z value -3.44 is in the critical region to the left (Hint: p<0.33) of -2.33
A Type I error is the mistake of ________ when it is actually true.
rejecting the null hypothesis
